Problem 25 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Slope \(3 ; \quad y\) intercept \(-2\)

Step-by-Step Solution

Answer

The equation is \( y = 3x - 2 \).

Step 1: Identify the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.

Step 2: Substitute the Given Slope

Substitute the given slope of \( 3 \) into the equation: \( y = 3x + b \).

Step 3: Substitute the Y-Intercept

Substitute the given y-intercept of \( -2 \) into the equation: \( y = 3x - 2 \).

Step 4: Write the Final Equation

The equation of the line with a slope of \( 3 \) and a y-intercept of \( -2 \) is \( y = 3x - 2 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express the equation of a straight line. It is one of the most commonly used formats for equations of lines in mathematics. The general form of this equation is given by:\[ y = mx + b \]where:

• \( y \) is the dependent variable or the output value on the y-axis.
• \( m \) stands for the slope of the line, which determines its steepness.
• \( x \) is the independent variable, representing the input value on the x-axis.
• \( b \) is the y-intercept, which indicates the point where the line crosses the y-axis.

The slope-intercept form makes it easy to identify both the slope and the y-intercept of a line just by looking at the equation. This form is particularly useful when plotting a line on a graph or when solving problems involving linear equations.

Slope

The slope is a measure of how steep a line is. It essentially tells you how much the line rises or falls as you move from one point to another along the line. The slope is denoted by the letter \( m \) in the slope-intercept form equation \( y = mx + b \).
When calculating the slope from two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line, you use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1}\]A slope of:

• \( m > 0 \) means the line is rising as it moves from left to right.
• \( m < 0 \) indicates the line is falling as it moves from left to right.
• \( m = 0 \) represents a horizontal line, which does not rise or fall.
• An undefined slope is associated with a vertical line.

In the example equation \( y = 3x - 2 \), the slope \( m \) is \( 3 \), meaning the line rises 3 units for every 1 unit it moves to the right.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis on a graph. This is always expressed as some value \( b \) when you write the line in slope-intercept form (\( y = mx + b \)). It's important because:

• The y-intercept gives you a starting point from which you can draw the line.
• This point can be easily identified as \((0, b)\) since at the y-intercept, the value of \( x \) is \( 0 \).

In the example provided, \( y = 3x - 2 \), the y-intercept \( b \) is \(-2\). This tells us that the line crosses the y-axis at the point \((0, -2)\). Understanding the y-intercept is crucial for graphing lines and solving real-world problems involving linear equations.